An Extension Theorem for convex functions of class $C^{1,1}$ on Hilbert spaces

TL;DR

This paper establishes a necessary and sufficient condition for extending functions to convex, $C^{1,1}$ class functions on Hilbert spaces, with applications to convex body interpolation and a counterexample in smooth convex extension theory.

Contribution

It provides a new extension theorem characterizing when a pair of functions can be extended to a convex $C^{1,1}$ function on Hilbert spaces, including a geometric application.

Findings

Characterization of conditions for convex $C^{1,1}$ extension on Hilbert spaces.

Extension can be achieved with the same Lipschitz constant for the gradient.

Counterexample to smooth convex extension with non-uniformly continuous derivatives.

Abstract

Let $\mathbb{H}$ be a Hilbert space, $E \subset \mathbb{H}$ be an arbitrary subset and $f: E \rightarrow \mathbb{R}, \: G: E \rightarrow \mathbb{H}$ be two functions. We give a necessary and sufficient condition on the pair $(f,G)$ for the existence of a \textit{convex} function $F\in C^{1,1}(\mathbb{H})$ such that $F=f$ and $\nabla F =G$ on $E$. We also show that, if this condition is met, $F$ can be taken so that $\textrm{Lip}(\nabla F) = \textrm{Lip}(G)$. We give a geometrical application of this result, concerning interpolation of sets by boundaries of $C^{1,1}$ convex bodies in $\mathbb{H}$. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.